Let $ \phi = \phi(x) $, $ \mathbf{u} = \mathbf{u}(x) $, and $ \mathbf{T} = \mathbf{T}(x) $ be differentiable scalar, vector, and tensor fields, where $ \mathbf{x} $ is the position vector. Show that The image contains the formula for the divergence of a product of a scalar field $ \phi $ and a vector field $ \mathbf{u} $:\[ \text{div} (\phi \mathbf{u}) = \mathbf{u} \cdot \nabla \phi + \phi \, \text{div} \, \mathbf{u} \]**Explanation:**- **$\text{div} (\phi \mathbf{u})$**: Represents the divergence of the product of $ \phi $ (a scalar field) and $ \mathbf{u} $ (a vector field).- **$\mathbf{u} \cdot \nabla \phi$**: Represents the dot product of the vector field $ \mathbf{u} $ and the gradient of the scalar field $ \phi $.- **$\phi \, \text{div} \, \mathbf{u}$**: Represents the product of the scalar field $ \phi $ and the divergence of the vector field $ \mathbf{u} $.This identity is useful in vector calculus when dealing with fluid dynamics, electromagnetism, and other fields involving differential equations.

Answered: div (øu) = u · grad o + o div u

Math

Advanced Math

div (øu) = u · grad o + o div u

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let $ \phi = \phi(x) $, $ \mathbf{u} = \mathbf{u}(x) $, and $ \mathbf{T} = \mathbf{T}(x) $ be differentiable scalar, vector, and tensor fields, where $ \mathbf{x} $ is the position vector. Show that

The image contains the formula for the divergence of a product of a scalar field $ \phi $ and a vector field $ \mathbf{u} $:

\[ \text{div} (\phi \mathbf{u}) = \mathbf{u} \cdot \nabla \phi + \phi \, \text{div} \, \mathbf{u} \]

**Explanation:**

- **$\text{div} (\phi \mathbf{u})$**: Represents the divergence of the product of $ \phi $ (a scalar field) and $ \mathbf{u} $ (a vector field).

- **$\mathbf{u} \cdot \nabla \phi$**: Represents the dot product of the vector field $ \mathbf{u} $ and the gradient of the scalar field $ \phi $.

- **$\phi \, \text{div} \, \mathbf{u}$**: Represents the product of the scalar field $ \phi $ and the divergence of the vector field $ \mathbf{u} $.

This identity is useful in vector calculus when dealing with fluid dynamics, electromagnetism, and other fields involving differential equations.

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